On the Hyers-Ulam Stability of Differential Equations of Second Order (original) (raw)

Abstract

By using of the Gronwall inequality, we prove the Hyers-Ulam stability of differential equations of second order with initial conditions. ∞ 0 | ( )| = 0 and = inf ≥0 ( ) = .

FAQs

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What are the sufficient conditions for Hyers-Ulam stability in differential equations?add

The study provides sufficient conditions ensuring that every solution of certain linear differential equations remains bounded, significantly impacting stability analyses.

How does the Gronwall lemma contribute to proving stability in this context?add

The paper employs the Gronwall lemma as a foundational method, effectively demonstrating the Hyers-Ulam stability of linear differential equations.

What specific results extend earlier findings on linear differential equations of second order?add

Findings extend known results, such as those by Alsina and Ger (2001), and generalize Hyers-Ulam stability to include second-order linear differential equations.

What numerical bounds were established for solutions of these differential equations?add

The theorem concludes that solutions are bounded by a constant proportional to the perturbation, ensuring stability quantitatively.

Which methods were utilized for analyzing Hyers-Ulam stability in nonlinear equations?add

The paper utilizes integral factors and applies advanced techniques to extend stability analysis to nonlinear differential equations effectively.

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References (25)

  1. S. M. Ulam, Problems in Modern Mathematics, Science Editions, Chapter 6, Wiley, New York, NY, USA, 1960.
  2. D. H. Hyers, "On the stability of the linear functional equation, " Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941.
  3. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Singapore, 2002.
  4. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, 1998.
  5. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equa- tions in Nonlinear Analysis, Springer, New York, NY, USA, 2011.
  6. T. M. Rassias, "On the stability of the linear mapping in Banach spaces, " Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978.
  7. M. Obloza, "Hyers stability of the linear differential equation, " Rocznik Naukowo-Dydaktyczny Prace Matematyczne, vol. 13, pp. 259-270, 1993.
  8. M. Obłoza, "Connections between Hyers and Lyapunov stabil- ity of the ordinary differential equations, " Rocznik Naukowo- Dydaktyczny. Prace Matematyczne, no. 14, pp. 141-146, 1997.
  9. C. Alsina and R. Ger, "On some inequalities and stability results related to the exponential function, " Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373-380, 1998.
  10. T. Miura, S. Miyajima, and S. E. Takahasi, "A characterization of Hyers-Ulam stability of first order linear differential operators, " Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136-146, 2003.
  11. S. E. Takahasi, T. Miura, and S. Miyajima, "On the Hyers-Ulam stability of the Banach space-valued differential equation 𝑦󸀠 = 𝜆𝑦, " Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309-315, 2002.
  12. S. Takahasi, H. Takagi, T. Miura, and S. Miyajima, "The Hyers-Ulam stability constants of first order linear differential operators, " Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 403-409, 2004.
  13. T. Miura, S. Miyajima, and S. Takahasi, "Hyers-Ulam stability of linear differential operator with constant coefficients, " Math- ematische Nachrichten, vol. 258, pp. 90-96, 2003.
  14. S.-M. Jung, "Hyers-Ulam stability of linear differential equa- tions of first order, " Applied Mathematics Letters, vol. 17, no. 10, pp. 1135-1140, 2004.
  15. S. M. Jung, "Hyers-Ulam stability of linear differential equations of first order,III, " Journal of Mathematical Analysis and Applica- tions, vol. 311, no. 1, pp. 139-146, 2005.
  16. S.-M. Jung, "Hyers-Ulam stability of linear differential equa- tions of first order, II, " Applied Mathematics Letters, vol. 19, no. 9, pp. 854-858, 2006.
  17. S. M. Jung, "Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, " Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 549-561, 2006.
  18. I. A. Rus, "Remarks on Ulam stability of the operatorial equa- tions, " Fixed Point Theory, vol. 10, no. 2, pp. 305-320, 2009.
  19. I. A. Rus, "Ulam stability of ordinary differential equations, " Studia Universitatis Babes-Bolyai: Mathematica, vol. 54, no. 4, pp. 125-133, 2009.
  20. Y. Li and Y. Shen, "Hyers-Ulam stability of linear differential equations of second order, " Applied Mathematics Letters, vol. 23, no. 3, pp. 306-309, 2010.
  21. G. Wang, M. Zhou, and L. Sun, "Hyers-Ulam stability of lin- ear differential equations of first order, " Applied Mathematics Letters, vol. 21, no. 10, pp. 1024-1028, 2008.
  22. D. S. Cimpean and D. Popa, "On the stability of the linear differential equation of higher order with constant coefficients, " Applied Mathematics and Computation, vol. 217, no. 8, pp. 4141- 4146, 2010.
  23. D. Popa and I. Ras ¸a, "On the Hyers-Ulam stability of the linear differential equation, " Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 530-537, 2011.
  24. D. Popa and I. Ras ¸a, "Hyers-Ulam stability of the linear differential operator with nonconstant coefficients, " Applied Mathematics and Computation, vol. 219, no. 4, pp. 1562-1568, 2012.
  25. H. Rezaei, S.-M. Jung, and T. M. Rassias, "Laplace transform and Hyers-Ulam stability of linear differential equations, " Journal of Mathematical Analysis and Applications, vol. 403, no. 1, pp. 244- 251, 2013.